"He's still not ready to play," Twins manager Ron Gardenhire said before the series opener against the Detroit Tigers.
Morneau was expected to be evaluated again Friday by the team's medical staff, but the first baseman will ultimately be the one who decides when he returns.
"He has to be the one who can tell us what he can and cannot do," Gardenhire said.
Morneau was selected to play in the All-Star Game and his status for it is uncertain.
"If he misses the whole series, the doctors and general manager will talk about what's the right thing," said Gardenhire, adding it wouldn't "make much sense" for him to play if he misses the series against the Tigers. "That's something we'll look at after Sunday."
"It's a good thing to be able to plug Thome into the four hole," Gardenhire said.
But Thome -- along with almost every other player in baseball -- can't duplicate what Morneau does at the plate.
The four-time All-Star has an American League-best .437 on-base percentage and is among the leaders with a .345 batting average, 18 homers, 56 RBIs and 25 doubles.
Morneau went on the disabled list four years ago after he was hit in the head by a pitch from the Seattle Mariners' Ron Villone and had headaches for several days, leading to him being out for 13 games. The Canadian also had head injuries during his days as a youth hockey goaltender and high school basketball player.